Division by Zero: Indeterminate or Undefined?

Date: 02/23/2002 at 16:21:24
From: Brant Langer Gurganus
Subject: My theories on Zero

In a previous response on a related topic, you said that division is 
defined in terms of reverse multiplication, and multiplication is 
defined in terms of repeated addition.  Therefore, division is
defined in terms of repeated subtraction.

If so, then division is a count of how many times a number may be
subtracted from another number until the original number equals zero.

Case 1: 0/X; X is not 0
  Operation:        Count:
  0                 0 (it's already at 0 so 0/x=0)
  0-X = -X          (past 0)

Case 2: X/0; X is not 0
  Operation:        Count:
  X                 0
  X-0 = X           1
  X-0 = X           2
   .                 .
   .                 .
   .                 .
  X-0 = X           infinity (not undefined)

Case 3: 0/0
  Operation:        Count:
  0                 0
  0-0 = 0           1
   .                 .
   .                 .
   .                 .
  0-0 = 0           infinity (0/0 can have all numbers 
                    as solutions)

Based on my cases above:

  1. 0/x = 0

  2. x/0 = infinity

  3. 0/0 = one or more elements of the set of all numbers 
     (depending on context)

Can you comment on these conclusions? 

Date: 02/23/2002 at 17:41:35
From: Doctor Rick
Subject: Re: My theories on Zero

Hi, Brant.

Your analysis seems like a reasonable way to describe and explain the 
results of dividing by zero. The only problem I have is that your use 
of the word "infinity" for both "infinitely large (greater than any 
number)" (what we call "undefined") and "can have any number as a 
solution" (what we call "indeterminate") is inconsistent and therefore 
confusing. See our explanations here:

  Dividing by 0 - Dr. Math FAQ
  http://mathforum.org/dr.math/faq/faq.divideby0.html   

  The indeterminate nature of 0/0
  http://mathforum.org/dr.math/problems/rob.12.21.00.html   

  Defining 0/0
  http://mathforum.org/dr.math/problems/howard.01.29.01.html   

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/23/2002 at 19:04:08
From: Brant Langer Gurganus
Subject: My theories on Zero

What I meant was that since no matter how many times 0 is subtracted 
from 0, it will always be 0; so (expanding beyond integer division) 
all numbers can be a solution to 0/0. The "one or more solutions" 
depends on the context of the problem.

Thanks for the quick reply.

Date: 02/23/2002 at 21:08:29
From: Doctor Rick
Subject: Re: My theories on Zero

Hi, Brant.

I know what you meant by "one or more solutions," and you're right. I 
hope you read the archive items I listed, which clarify how this works 
out in practice. 

As I put it, "indeterminate" means that there may be a definite 
answer to the problem you are working on, but you'll have to back up 
and find another way to discover it; dividing zero by zero cannot tell 
you the answer. It's sort of a "road closed" sign.

A similar statement applies to "undefined." Rather than say that it 
takes an infinite number of subtractions of zero to get x down to 0, 
we have to say that NO number of subtractions of zero will get x down 
to 0. The answer to x/0 is not a number. That's another good way to 
say "undefined," and some computer languages call it this.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/